Ising model susceptibility: The Fuchsian differential equation for χ⁽⁴⁾ and its factorization properties

We give the Fuchsian linear differential equation satisfied by χ (4), the 'four-particle' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to χ(4). The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution χ (2) is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, χ(3), leads us to a conjecture on the structure of all the n-particle contributions χ(n). © 2005 IOP Publishing Ltd.